On the Local Linear Rate of Consensus on the Stiefel Manifold
On the Local Linear Rate of Consensus on the Stiefel Manifold
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DOI：
10.1109/tac.2023.3330735
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发表时间：
2021-01
影响因子：
6.8
通讯作者：
Shixiang Chen;Alfredo García;Mingyi Hong;Shahin Shahrampour
中科院分区：
文献类型：
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作者：
Shixiang Chen;Alfredo García;Mingyi Hong;Shahin Shahrampour
由纯粹的局部交互产生的协调群体行为已经成功地建立了分布式共识寻求动力学模型，其中局部行为的目标是最小化与相邻同伴的分歧。然而，最近的研究表明，当受到流形几何的约束时，分布式寻求共识的动态可能最终无法收敛到全局共识状态。在本文中，我们研究了Stiefel流形上的离散时间寻求共识动力学，并确定了网络拓扑上确保收敛到全局共识状态的条件。我们进一步证明了共识状态的（局部）线性收敛速度与欧氏空间中已知的速度相当。这些结果对受流形几何约束的一致性应用具有启示意义，例如同步和集体运动，并且它们可用于Stiefel流形上分散黎曼优化的收敛分析。
Coordinated group behavior arising from purely local interactions has been successfully modeled with distributed consensus-seeking dynamics, where the local behavior is aimed at minimizing the disagreement with neighboring peers. However, it has been recently shown that when constrained by a manifold geometry, distributed consensus-seeking dynamics may ultimately fail to converge to a global consensus state. In this article, we study discrete-time consensus-seeking dynamics on the Stiefel manifold and identify conditions on the network topology to ensure convergence to a global consensus state. We further prove a (local) linear convergence rate to the consensus state that is on par with the well-known rate in the Euclidean space. These results have implications for consensus applications constrained by manifold geometry, such as synchronization and collective motion, and they can be used for convergence analysis of decentralized Riemannian optimization on the Stiefel Manifold.